A234095 -id:A234095 - cp's OEIS frontend

A234093 Integers of the form (p*q - 1)/2, where p and q are distinct primes.

7, 10, 16, 17, 19, 25, 27, 28, 32, 34, 38, 42, 43, 45, 46, 47, 55, 57, 59, 61, 64, 66, 70, 71, 72, 77, 79, 80, 88, 91, 92, 93, 100, 101, 102, 104, 106, 107, 108, 109, 110, 117, 118, 123, 124, 126, 129, 132, 133, 143, 145, 147, 149, 150, 151, 152, 154, 159

Offset: 1

Clark Kimberling, Dec 27 2013

A102770 is subsequence. - Zak Seidov, Feb 21 2014

			(3*5 - 1)/2 = 7; (3*7 - 1)/2 = 10.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A102770, A233561, A234095, A234096.

t = Select[Range[1, 7000, 2], Map[Last, FactorInteger[#]] == Table[1, {2}] &]; Take[(t - 1)/2, 120] (* A234093 *)
v = Flatten[Position[PrimeQ[(t - 1)/2], True]] ; w = Table[t[[v[[n]]]], {n, 1, Length[v]}]  (* A233561 *)
(w - 1)/2 (* A234095 *) (* Peter J. C. Moses, Dec 23 2013 *)
With[{nn=60},Take[Union[Select[(Times@@#-1)/2&/@Subsets[Prime[ Range[ nn]],{2}],IntegerQ]],nn]] (* Harvey P. Dale, Mar 08 2015 *)

A286258 Compound filter: a(n) = P(A046523(n), A046523(2n+1)), where P(n,k) is sequence A000027 used as a pairing function.

Original entry on oeis.org

2, 5, 5, 25, 5, 27, 23, 44, 14, 61, 5, 117, 38, 27, 27, 226, 23, 90, 23, 90, 27, 142, 5, 375, 40, 27, 86, 148, 5, 495, 80, 698, 27, 61, 27, 702, 80, 61, 27, 765, 5, 625, 23, 90, 148, 61, 23, 1224, 109, 90, 27, 832, 5, 324, 61, 324, 61, 142, 23, 2013, 23, 84, 90, 2410, 27, 625, 302, 90, 27, 625, 23, 2998, 80, 27, 90, 265, 61, 495, 23, 1426, 152, 601, 5, 2013, 142, 27, 142

Offset: 1

Antti Karttunen, May 07 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pairing Function

Cf. A000027, A046523, A286255, A286256, A286257.

Cf. A005384 (gives the positions of 5's), A234095 (of 23's).

A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ This function from Charles R Greathouse IV, Aug 17 2011
A286258(n) = (1/2)*(2 + ((A046523(n)+A046523((2*n)+1))^2) - A046523(n) - 3*A046523((2*n)+1));
for(n=1, 10000, write("b286258.txt", n, " ", A286258(n)));

from sympy import factorint
def T(n, m): return ((n + m)**2 - n - 3*m + 2)/2
def P(n):
    f = factorint(n)
    return sorted([f[i] for i in f])
def a046523(n):
    x=1
    while True:
        if P(n) == P(x): return x
        else: x+=1
def a(n): return T(a046523(n), a046523(2*n + 1)) # Indranil Ghosh, May 07 2017

(define (A286258 n) (* (/ 1 2) (+ (expt (+ (A046523 n) (A046523 (+ 1 n n))) 2) (- (A046523 n)) (- (* 3 (A046523 (+ 1 n n)))) 2)))

a(n) = (1/2)*(2 + ((A046523(n)+A046523((2*n)+1))^2) - A046523(n) - 3*A046523((2*n)+1)).

A233561 Products pq of distinct primes such that (pq - 1)/2 is prime.

Original entry on oeis.org

15, 35, 39, 87, 95, 119, 123, 143, 159, 203, 215, 219, 299, 303, 327, 335, 395, 447, 515, 527, 543, 623, 635, 695, 699, 707, 767, 779, 803, 843, 879, 899, 923, 959, 1007, 1043, 1047, 1115, 1139, 1199, 1203, 1227, 1263, 1347, 1355, 1383, 1403, 1623, 1643

Offset: 1

Clark Kimberling, Dec 14 2013

			15 = 3*5 is the least product of distinct primes p and q for which (p*q - 1)/2 is prime, so a(1) = 15.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A234093, A234095, A046388.

t = Select[Range[1, 7000, 2], Map[Last, FactorInteger[#]] == Table[1, {2}] &]; Take[(t - 1)/2, 120] (* A234093 *)
v = Flatten[Position[PrimeQ[(t - 1)/2], True]] ; w = Table[t[[v[[n]]]], {n, 1, Length[v]}]  (* A233561 *)
(w - 1)/2 (* A234095 *)  (* Peter J. C. Moses, Dec 23 2013 *)
With[{upto=2000},Select[Times@@#&/@Select[Subsets[Prime[Range[ PrimePi[ upto/2]]],{2}],PrimeQ[(Times@@#-1)/2]&]//Union,#<=upto&]] (* Harvey P. Dale, Nov 02 2017 *)

A305978 Filter sequence combining prime signatures of n and 2n+1.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 5, 6, 7, 8, 2, 9, 10, 4, 4, 11, 5, 12, 5, 12, 4, 13, 2, 14, 15, 4, 16, 17, 2, 18, 19, 20, 4, 8, 4, 21, 19, 8, 4, 22, 2, 23, 5, 12, 17, 8, 5, 24, 25, 12, 4, 26, 2, 27, 8, 27, 8, 13, 5, 28, 5, 29, 12, 30, 4, 23, 31, 12, 4, 23, 5, 32, 19, 4, 12, 33, 8, 18, 5, 34, 35, 36, 2, 28, 13, 4, 13, 37, 2, 38, 8, 17, 8, 39, 4, 40, 41, 12, 12, 42, 5, 23

Offset: 1

Antti Karttunen, Jun 15 2018

nonn

Restricted growth sequence transform of A286258.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A046523, A101296, A286258, A305810, A305973, A305977.

Cf. A005384 (positions of 2's), A234095 (of 5's).

up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523
Aux305978(n) = [A046523(n),A046523(n+n+1)];
v305978 = rgs_transform(vector(up_to,n,Aux305978(n)));
A305978(n) = v305978[n];

A234098 Primes of the form (p*q + 1)/2, where p and q are distinct primes.

Original entry on oeis.org

11, 17, 29, 43, 47, 67, 71, 73, 89, 101, 103, 107, 109, 127, 151, 191, 197, 223, 227, 241, 251, 269, 277, 283, 317, 349, 359, 373, 397, 409, 433, 457, 461, 467, 487, 521, 541, 569, 571, 631, 643, 647, 659, 673, 701, 709, 719, 733, 739, 751, 757, 769, 821

Offset: 1

Clark Kimberling, Dec 27 2013

			11 = (3*7 + 1)/2, 17 = (5*7 + 1)/2.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A234096, A233562, A234095.

Cf. A010051, A046388.

a234098 n = a234098_list !! (n-1)
a234098_list = filter ((== 1) . a010051') $
                      map ((flip div 2) . (+ 1)) a046388_list
-- Reinhard Zumkeller, Jan 02 2014

t = Select[Range[1, 7000, 2], Map[Last, FactorInteger[#]] == Table[1, {2}] &]; Take[(t + 1)/2, 120] (* A234096 *)
v = Flatten[Position[PrimeQ[(t + 1)/2], True]] ; w = Table[t[[v[[n]]]], {n, 1, Length[v]}]  (* A233562 *)
(w + 1)/2 (* A234098 *)  (* Peter J. C. Moses, Dec 23 2013 *)
Take[Select[(Times@@#+1)/2&/@Subsets[Prime[Range[200]],{2}],PrimeQ]//Union,60] (* Harvey P. Dale, Jun 24 2025 *)

A319706 Filter sequence which for primes p records the prime signature of 2p+1, and for all other numbers assigns a unique number.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 5, 6, 7, 8, 2, 9, 10, 11, 12, 13, 5, 14, 5, 15, 16, 17, 2, 18, 19, 20, 21, 22, 2, 23, 24, 25, 26, 27, 28, 29, 24, 30, 31, 32, 2, 33, 5, 34, 35, 36, 5, 37, 38, 39, 40, 41, 2, 42, 43, 44, 45, 46, 5, 47, 5, 48, 49, 50, 51, 52, 53, 54, 55, 56, 5, 57, 24, 58, 59, 60, 61, 62, 5, 63, 64, 65, 2, 66, 67, 68, 69, 70, 2, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 5

Offset: 1

Antti Karttunen, Sep 26 2018

nonn

Restricted growth sequence transform of function f defined as f(n) = A046523(2n+1) when n is a prime, otherwise -n.
For all i, j:
  A305810(i) = A305810(j) => a(i) = a(j),
and
  a(i) = a(j) => A305800(i) = A305800(j),
  a(i) = a(j) => A305978(i) = A305978(j),
  a(i) = a(j) => A305985(i) = A305985(j).

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf- A046523, A305800, A305810, A305900, A305901, A305978, A305985.

Cf. A005384 (positions of 2's), A234095 (positions of 5's).

up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
A319706aux(n) = if(isprime(n),A046523(n+n+1),-n);
v319706 = rgs_transform(vector(up_to,n,A319706aux(n)));
A319706(n) = v319706[n];

A276045 Primes p such that d(p*(2p+1)) = 8 where d(n) is the number of divisors of n (A000005).

Original entry on oeis.org

7, 13, 17, 19, 43, 47, 59, 61, 71, 79, 101, 107, 109, 149, 151, 163, 167, 197, 223, 257, 263, 271, 311, 317, 347, 349, 353, 383, 389, 401, 421, 439, 449, 461, 479, 503, 521, 523, 557, 569, 599, 601, 613, 631, 673, 677, 691, 701, 811, 821, 827, 839, 853, 863, 881, 919

Offset: 1

Anthony Hernandez, Aug 17 2016

Primes p such that 2p+1 is in A030513. - Robert Israel, Aug 17 2016
From Anthony Hernandez, Aug 29 2016: (Start)
Conjecture: this sequence is infinite.
It appears that the prime numbers in this sequence which have 7 for as final digit form the sequence A104164.
Conjecture: this sequence contains infinitely many twin primes. The first few twin primes in this sequence are 17,19,59,61,107,109,521,523,599,601,... (End)
From Bernard Schott, Apr 28 2020: (Start)
This sequence equals the union of {13} and A234095; proof by double inclusion:
-> 1st inclusion: {13} Union A234095 is included in A276045.
1) if p = 13, then 13*27 = 351 = 3^3 * 13, hence d(351) = 8 and 13 belongs to A276045.
2) if p is in A234095, then p*(2*p+1) = p*r*s (p,r,s primes) and d(p*r*s) = 8, hence p is in 276045.
-> 2nd inclusion: A276045 is included in {13} Union A234095.
If p is in A276095, then m=p*(2*p+1) has 8 divisors and there are only three possibilities: m = u*v*w, or m = u^3*v or m = u^7 with u, v, w are distinct primes.
1st case: if p*(2*p+1) = u*v*w then u=p, and 2p+1=v*w is semiprime; hence, p is in A234095 Union {13}.
2nd case: if p*(2p+1) = u^3*v then  p=v and 2*p+1=u^3 ==> 2*p = u^3-1 = (u-1)*(u^2+u+1) with 2 and p are primes; then (u-1=2, u^2+u+1=p) so u=3, and p=3^2+3+1=13; hence p = 13 belongs to {13} Union A234095.
3rd case: p*(2p+1) = u^7 is impossible.
Conclusion: this sequence = {13} Union A234095. (End)

			d(7*(2*7+1))=d(105)=8 so 7 is a term.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A030513, A030626.

Equals {13} Union A234095.

select(n -> isprime(n) and numtheory:-tau(n*(2*n+1))=8,
[seq(i, i=3..1000, 2)]); # Robert Israel, Aug 17 2016

Select[Prime@ Range@ 160, DivisorSigma[0, # (2 # + 1)] == 8 &] (* Michael De Vlieger, Aug 28 2016 *)

lista(nn) = forprime(p=2, nn, if (numdiv(p*(2*p+1))==8, print1(p, ", "))); \\ Michel Marcus, Aug 17 2016

A235646 Primes p such that b=2p+1 is semiprime, c=2b+1 is 3-almost prime and d=2*c+1 is 4-almost prime.

Original entry on oeis.org

43, 1429, 2239, 3319, 4831, 6379, 8821, 10501, 11383, 12781, 13003, 14771, 15091, 16063, 16759, 18223, 19213, 19681, 20021, 22571, 24103, 24109, 24571, 25939, 27271, 28933, 29833, 30241, 31723, 33679, 33811, 34381, 34781, 35591, 35863, 39373

Offset: 1

Zak Seidov, Feb 20 2014

nonn

			p = 43, b = 87 = 2*29 = A001358(30), c = 175 = 5*5*7 = A014612(43), d = 351 = 3*3*3**13 =  A014613(50).

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Subsequence of A238202.

Cf. A000040, A001358, A014612, A014613, A234095.

Select[ Prime@ Range@ 4200,PrimeOmega[2#+1]==2 && PrimeOmega[4#+3]==3 && PrimeOmega[8#+7]==4 &] (* Robert G. Wilson v, Feb 22 2014 *)

forprime(p=43,40000,if(bigomega(a=2*p+1)==2&& bigomega(b=2*c+1)==3&&bigomega(c=2*d+1)==4,print(p",")))

A238202 Primes p such that b = 2p + 1 is semiprime and c = 2b + 1 is 3-almost prime.

Original entry on oeis.org

43, 503, 631, 827, 971, 1063, 1153, 1283, 1373, 1429, 1433, 1493, 1523, 1553, 1619, 1693, 1877, 2113, 2239, 2243, 2297, 2423, 2477, 2531, 2609, 2683, 2851, 2927, 2999, 3203, 3221, 3319, 3463, 3533, 3557, 3571, 3583, 3701

Offset: 1

Zak Seidov, Feb 20 2014

nonn

Subsequence of A234095 and A000040, e.g., a(1) = 43 = A234095(4) = A000040(14).

			2*43 + 1 = 87 = 2*29 = A001358(30), 2*87 + 1 = 175 = 5*5*7 = A014612(43).

Cf. A000040, A001358, A014612, A234095.

forprime(p=2,4000,if(bigomega(a=2*p+1)==2&&bigomega(b=2*a+1)==3,print(p",")))

A328058 Primes p such that 2*p-1 is a semiprime.

Original entry on oeis.org

5, 11, 13, 17, 29, 43, 47, 61, 67, 71, 73, 89, 101, 103, 107, 109, 127, 151, 181, 191, 197, 223, 227, 241, 251, 269, 277, 283, 317, 349, 359, 373, 397, 409, 421, 433, 457, 461, 467, 487, 521, 541, 569, 571, 631, 643, 647, 659, 673, 701, 709, 719, 733, 739, 751, 757, 769, 821, 857, 859, 881, 883

Offset: 1

J. M. Bergot and Robert Israel, Oct 03 2019

nonn

			a(3)=13 is in the sequence because it is prime and 2*13-1=5^2 is a semiprime.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A001358. Includes A067756 and A162336.

Cf. A086006, A234095.

[p: p in PrimesUpTo(1000)| &+[d[2]: d in Factorization(2*p-1)] eq 2]; // Marius A. Burtea, Oct 03 2019

select(t -> isprime(t) and numtheory:-bigomega(2*t-1)=2, [2,seq(i,i=3..10000,2)]);

Select[Prime@ Range@ 153, PrimeOmega[2 # - 1] == 2 &] (* Michael De Vlieger, Oct 03 2019 *)

isok(p) = isprime(p) && (bigomega(2*p-1) == 2); \\ Michel Marcus, Oct 04 2019

cp's OEIS Frontend

A234093 Integers of the form (p*q - 1)/2, where p and q are distinct primes.

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A286258 Compound filter: a(n) = P(A046523(n), A046523(2n+1)), where P(n,k) is sequence A000027 used as a pairing function.

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A233561 Products p*q of distinct primes such that (p*q - 1)/2 is prime.

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Mathematica

A305978 Filter sequence combining prime signatures of n and 2n+1.

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A234098 Primes of the form (p*q + 1)/2, where p and q are distinct primes.

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Haskell

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A319706 Filter sequence which for primes p records the prime signature of 2p+1, and for all other numbers assigns a unique number.

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A276045 Primes p such that d(p*(2p+1)) = 8 where d(n) is the number of divisors of n (A000005).

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A235646 Primes p such that b=2*p+1 is semiprime, c=2*b+1 is 3-almost prime and d=2*c+1 is 4-almost prime.

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A233561 Products pq of distinct primes such that (pq - 1)/2 is prime.

A235646 Primes p such that b=2p+1 is semiprime, c=2b+1 is 3-almost prime and d=2*c+1 is 4-almost prime.

A238202 Primes p such that b = 2p + 1 is semiprime and c = 2b + 1 is 3-almost prime.